On the cells in a stationary Poisson hyperplane mosaic

Abstract

Let $X$ be the mosaic generated by a stationary Poisson hyperplane process $\hat{X}$ in ${\mathbb{R}}^d$. Under some mild conditions on the spherical directional distribution of $\hat{X}$ (which are satisfied, for example, if the process is isotropic), we show that with probability one the set of cells ($d$-polytopes) of $X$ has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple $d$-polytopes is realized infinitely often by the cells of $X$. A further result concerns the distribution of the typical cell.